# Properties

 Label 1680.2.t.h.1009.4 Level $1680$ Weight $2$ Character 1680.1009 Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.t (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1009.4 Root $$1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 1680.1009 Dual form 1680.2.t.h.1009.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +2.23607 q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +2.23607 q^{5} +1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -4.47214i q^{13} +2.23607i q^{15} -6.47214i q^{17} +2.00000 q^{19} -1.00000 q^{21} -4.00000i q^{23} +5.00000 q^{25} -1.00000i q^{27} +8.47214 q^{29} +0.472136 q^{31} +2.00000i q^{33} +2.23607i q^{35} -2.47214i q^{37} +4.47214 q^{39} -3.52786 q^{41} +2.47214i q^{43} -2.23607 q^{45} +6.47214i q^{47} -1.00000 q^{49} +6.47214 q^{51} +2.00000i q^{53} +4.47214 q^{55} +2.00000i q^{57} -3.52786 q^{61} -1.00000i q^{63} -10.0000i q^{65} +1.52786i q^{67} +4.00000 q^{69} -12.4721 q^{71} +7.52786i q^{73} +5.00000i q^{75} +2.00000i q^{77} +8.94427 q^{79} +1.00000 q^{81} +4.94427i q^{83} -14.4721i q^{85} +8.47214i q^{87} +17.4164 q^{89} +4.47214 q^{91} +0.472136i q^{93} +4.47214 q^{95} -3.52786i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{9} + 8 q^{11} + 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} - 16 q^{31} - 32 q^{41} - 4 q^{49} + 8 q^{51} - 32 q^{61} + 16 q^{69} - 32 q^{71} + 4 q^{81} + 16 q^{89} - 8 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 2.23607 1.00000
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ − 4.47214i − 1.24035i −0.784465 0.620174i $$-0.787062\pi$$
0.784465 0.620174i $$-0.212938\pi$$
$$14$$ 0 0
$$15$$ 2.23607i 0.577350i
$$16$$ 0 0
$$17$$ − 6.47214i − 1.56972i −0.619671 0.784862i $$-0.712734\pi$$
0.619671 0.784862i $$-0.287266\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 8.47214 1.57324 0.786618 0.617440i $$-0.211830\pi$$
0.786618 + 0.617440i $$0.211830\pi$$
$$30$$ 0 0
$$31$$ 0.472136 0.0847981 0.0423991 0.999101i $$-0.486500\pi$$
0.0423991 + 0.999101i $$0.486500\pi$$
$$32$$ 0 0
$$33$$ 2.00000i 0.348155i
$$34$$ 0 0
$$35$$ 2.23607i 0.377964i
$$36$$ 0 0
$$37$$ − 2.47214i − 0.406417i −0.979136 0.203208i $$-0.934863\pi$$
0.979136 0.203208i $$-0.0651369\pi$$
$$38$$ 0 0
$$39$$ 4.47214 0.716115
$$40$$ 0 0
$$41$$ −3.52786 −0.550960 −0.275480 0.961307i $$-0.588837\pi$$
−0.275480 + 0.961307i $$0.588837\pi$$
$$42$$ 0 0
$$43$$ 2.47214i 0.376997i 0.982073 + 0.188499i $$0.0603621\pi$$
−0.982073 + 0.188499i $$0.939638\pi$$
$$44$$ 0 0
$$45$$ −2.23607 −0.333333
$$46$$ 0 0
$$47$$ 6.47214i 0.944058i 0.881583 + 0.472029i $$0.156478\pi$$
−0.881583 + 0.472029i $$0.843522\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 6.47214 0.906280
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 4.47214 0.603023
$$56$$ 0 0
$$57$$ 2.00000i 0.264906i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −3.52786 −0.451697 −0.225848 0.974162i $$-0.572515\pi$$
−0.225848 + 0.974162i $$0.572515\pi$$
$$62$$ 0 0
$$63$$ − 1.00000i − 0.125988i
$$64$$ 0 0
$$65$$ − 10.0000i − 1.24035i
$$66$$ 0 0
$$67$$ 1.52786i 0.186658i 0.995635 + 0.0933292i $$0.0297509\pi$$
−0.995635 + 0.0933292i $$0.970249\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −12.4721 −1.48017 −0.740085 0.672513i $$-0.765215\pi$$
−0.740085 + 0.672513i $$0.765215\pi$$
$$72$$ 0 0
$$73$$ 7.52786i 0.881070i 0.897735 + 0.440535i $$0.145211\pi$$
−0.897735 + 0.440535i $$0.854789\pi$$
$$74$$ 0 0
$$75$$ 5.00000i 0.577350i
$$76$$ 0 0
$$77$$ 2.00000i 0.227921i
$$78$$ 0 0
$$79$$ 8.94427 1.00631 0.503155 0.864196i $$-0.332173\pi$$
0.503155 + 0.864196i $$0.332173\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.94427i 0.542704i 0.962480 + 0.271352i $$0.0874708\pi$$
−0.962480 + 0.271352i $$0.912529\pi$$
$$84$$ 0 0
$$85$$ − 14.4721i − 1.56972i
$$86$$ 0 0
$$87$$ 8.47214i 0.908308i
$$88$$ 0 0
$$89$$ 17.4164 1.84614 0.923068 0.384637i $$-0.125673\pi$$
0.923068 + 0.384637i $$0.125673\pi$$
$$90$$ 0 0
$$91$$ 4.47214 0.468807
$$92$$ 0 0
$$93$$ 0.472136i 0.0489582i
$$94$$ 0 0
$$95$$ 4.47214 0.458831
$$96$$ 0 0
$$97$$ − 3.52786i − 0.358200i −0.983831 0.179100i $$-0.942681\pi$$
0.983831 0.179100i $$-0.0573186\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 4.47214 0.444994 0.222497 0.974933i $$-0.428579\pi$$
0.222497 + 0.974933i $$0.428579\pi$$
$$102$$ 0 0
$$103$$ − 12.9443i − 1.27544i −0.770270 0.637719i $$-0.779878\pi$$
0.770270 0.637719i $$-0.220122\pi$$
$$104$$ 0 0
$$105$$ −2.23607 −0.218218
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ −14.9443 −1.43140 −0.715701 0.698407i $$-0.753893\pi$$
−0.715701 + 0.698407i $$0.753893\pi$$
$$110$$ 0 0
$$111$$ 2.47214 0.234645
$$112$$ 0 0
$$113$$ 14.9443i 1.40584i 0.711269 + 0.702919i $$0.248121\pi$$
−0.711269 + 0.702919i $$0.751879\pi$$
$$114$$ 0 0
$$115$$ − 8.94427i − 0.834058i
$$116$$ 0 0
$$117$$ 4.47214i 0.413449i
$$118$$ 0 0
$$119$$ 6.47214 0.593300
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 3.52786i − 0.318097i
$$124$$ 0 0
$$125$$ 11.1803 1.00000
$$126$$ 0 0
$$127$$ 4.00000i 0.354943i 0.984126 + 0.177471i $$0.0567917\pi$$
−0.984126 + 0.177471i $$0.943208\pi$$
$$128$$ 0 0
$$129$$ −2.47214 −0.217659
$$130$$ 0 0
$$131$$ −13.8885 −1.21345 −0.606724 0.794913i $$-0.707517\pi$$
−0.606724 + 0.794913i $$0.707517\pi$$
$$132$$ 0 0
$$133$$ 2.00000i 0.173422i
$$134$$ 0 0
$$135$$ − 2.23607i − 0.192450i
$$136$$ 0 0
$$137$$ 19.8885i 1.69919i 0.527433 + 0.849596i $$0.323154\pi$$
−0.527433 + 0.849596i $$0.676846\pi$$
$$138$$ 0 0
$$139$$ −2.94427 −0.249730 −0.124865 0.992174i $$-0.539850\pi$$
−0.124865 + 0.992174i $$0.539850\pi$$
$$140$$ 0 0
$$141$$ −6.47214 −0.545052
$$142$$ 0 0
$$143$$ − 8.94427i − 0.747958i
$$144$$ 0 0
$$145$$ 18.9443 1.57324
$$146$$ 0 0
$$147$$ − 1.00000i − 0.0824786i
$$148$$ 0 0
$$149$$ 12.4721 1.02176 0.510879 0.859653i $$-0.329320\pi$$
0.510879 + 0.859653i $$0.329320\pi$$
$$150$$ 0 0
$$151$$ 17.8885 1.45575 0.727875 0.685710i $$-0.240508\pi$$
0.727875 + 0.685710i $$0.240508\pi$$
$$152$$ 0 0
$$153$$ 6.47214i 0.523241i
$$154$$ 0 0
$$155$$ 1.05573 0.0847981
$$156$$ 0 0
$$157$$ − 9.41641i − 0.751511i −0.926719 0.375756i $$-0.877383\pi$$
0.926719 0.375756i $$-0.122617\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ − 23.4164i − 1.83411i −0.398755 0.917057i $$-0.630558\pi$$
0.398755 0.917057i $$-0.369442\pi$$
$$164$$ 0 0
$$165$$ 4.47214i 0.348155i
$$166$$ 0 0
$$167$$ 10.4721i 0.810358i 0.914237 + 0.405179i $$0.132791\pi$$
−0.914237 + 0.405179i $$0.867209\pi$$
$$168$$ 0 0
$$169$$ −7.00000 −0.538462
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 0 0
$$175$$ 5.00000i 0.377964i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −22.9443 −1.71494 −0.857468 0.514538i $$-0.827964\pi$$
−0.857468 + 0.514538i $$0.827964\pi$$
$$180$$ 0 0
$$181$$ 16.4721 1.22436 0.612182 0.790717i $$-0.290292\pi$$
0.612182 + 0.790717i $$0.290292\pi$$
$$182$$ 0 0
$$183$$ − 3.52786i − 0.260787i
$$184$$ 0 0
$$185$$ − 5.52786i − 0.406417i
$$186$$ 0 0
$$187$$ − 12.9443i − 0.946579i
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 17.4164 1.26021 0.630104 0.776511i $$-0.283012\pi$$
0.630104 + 0.776511i $$0.283012\pi$$
$$192$$ 0 0
$$193$$ 12.9443i 0.931749i 0.884851 + 0.465875i $$0.154260\pi$$
−0.884851 + 0.465875i $$0.845740\pi$$
$$194$$ 0 0
$$195$$ 10.0000 0.716115
$$196$$ 0 0
$$197$$ 19.8885i 1.41700i 0.705711 + 0.708500i $$0.250628\pi$$
−0.705711 + 0.708500i $$0.749372\pi$$
$$198$$ 0 0
$$199$$ 20.4721 1.45123 0.725616 0.688100i $$-0.241555\pi$$
0.725616 + 0.688100i $$0.241555\pi$$
$$200$$ 0 0
$$201$$ −1.52786 −0.107767
$$202$$ 0 0
$$203$$ 8.47214i 0.594627i
$$204$$ 0 0
$$205$$ −7.88854 −0.550960
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −25.8885 −1.78224 −0.891120 0.453767i $$-0.850080\pi$$
−0.891120 + 0.453767i $$0.850080\pi$$
$$212$$ 0 0
$$213$$ − 12.4721i − 0.854577i
$$214$$ 0 0
$$215$$ 5.52786i 0.376997i
$$216$$ 0 0
$$217$$ 0.472136i 0.0320507i
$$218$$ 0 0
$$219$$ −7.52786 −0.508686
$$220$$ 0 0
$$221$$ −28.9443 −1.94700
$$222$$ 0 0
$$223$$ − 20.9443i − 1.40253i −0.712900 0.701266i $$-0.752618\pi$$
0.712900 0.701266i $$-0.247382\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ − 9.88854i − 0.656326i −0.944621 0.328163i $$-0.893571\pi$$
0.944621 0.328163i $$-0.106429\pi$$
$$228$$ 0 0
$$229$$ 26.3607 1.74196 0.870981 0.491316i $$-0.163484\pi$$
0.870981 + 0.491316i $$0.163484\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 0 0
$$233$$ − 2.94427i − 0.192886i −0.995339 0.0964428i $$-0.969254\pi$$
0.995339 0.0964428i $$-0.0307465\pi$$
$$234$$ 0 0
$$235$$ 14.4721i 0.944058i
$$236$$ 0 0
$$237$$ 8.94427i 0.580993i
$$238$$ 0 0
$$239$$ 15.5279 1.00441 0.502207 0.864747i $$-0.332522\pi$$
0.502207 + 0.864747i $$0.332522\pi$$
$$240$$ 0 0
$$241$$ −11.8885 −0.765808 −0.382904 0.923788i $$-0.625076\pi$$
−0.382904 + 0.923788i $$0.625076\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ −2.23607 −0.142857
$$246$$ 0 0
$$247$$ − 8.94427i − 0.569110i
$$248$$ 0 0
$$249$$ −4.94427 −0.313331
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ − 8.00000i − 0.502956i
$$254$$ 0 0
$$255$$ 14.4721 0.906280
$$256$$ 0 0
$$257$$ − 23.4164i − 1.46068i −0.683086 0.730338i $$-0.739363\pi$$
0.683086 0.730338i $$-0.260637\pi$$
$$258$$ 0 0
$$259$$ 2.47214 0.153611
$$260$$ 0 0
$$261$$ −8.47214 −0.524412
$$262$$ 0 0
$$263$$ − 4.00000i − 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ 0 0
$$265$$ 4.47214i 0.274721i
$$266$$ 0 0
$$267$$ 17.4164i 1.06587i
$$268$$ 0 0
$$269$$ −30.3607 −1.85112 −0.925562 0.378597i $$-0.876407\pi$$
−0.925562 + 0.378597i $$0.876407\pi$$
$$270$$ 0 0
$$271$$ −24.4721 −1.48658 −0.743288 0.668971i $$-0.766735\pi$$
−0.743288 + 0.668971i $$0.766735\pi$$
$$272$$ 0 0
$$273$$ 4.47214i 0.270666i
$$274$$ 0 0
$$275$$ 10.0000 0.603023
$$276$$ 0 0
$$277$$ − 23.4164i − 1.40696i −0.710717 0.703478i $$-0.751629\pi$$
0.710717 0.703478i $$-0.248371\pi$$
$$278$$ 0 0
$$279$$ −0.472136 −0.0282660
$$280$$ 0 0
$$281$$ 2.94427 0.175641 0.0878203 0.996136i $$-0.472010\pi$$
0.0878203 + 0.996136i $$0.472010\pi$$
$$282$$ 0 0
$$283$$ − 16.9443i − 1.00723i −0.863927 0.503616i $$-0.832003\pi$$
0.863927 0.503616i $$-0.167997\pi$$
$$284$$ 0 0
$$285$$ 4.47214i 0.264906i
$$286$$ 0 0
$$287$$ − 3.52786i − 0.208243i
$$288$$ 0 0
$$289$$ −24.8885 −1.46403
$$290$$ 0 0
$$291$$ 3.52786 0.206807
$$292$$ 0 0
$$293$$ 24.9443i 1.45726i 0.684908 + 0.728630i $$0.259843\pi$$
−0.684908 + 0.728630i $$0.740157\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2.00000i − 0.116052i
$$298$$ 0 0
$$299$$ −17.8885 −1.03452
$$300$$ 0 0
$$301$$ −2.47214 −0.142492
$$302$$ 0 0
$$303$$ 4.47214i 0.256917i
$$304$$ 0 0
$$305$$ −7.88854 −0.451697
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ 12.9443 0.736374
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ − 25.4164i − 1.43662i −0.695723 0.718310i $$-0.744916\pi$$
0.695723 0.718310i $$-0.255084\pi$$
$$314$$ 0 0
$$315$$ − 2.23607i − 0.125988i
$$316$$ 0 0
$$317$$ − 6.94427i − 0.390029i −0.980800 0.195015i $$-0.937525\pi$$
0.980800 0.195015i $$-0.0624754\pi$$
$$318$$ 0 0
$$319$$ 16.9443 0.948697
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 12.9443i − 0.720239i
$$324$$ 0 0
$$325$$ − 22.3607i − 1.24035i
$$326$$ 0 0
$$327$$ − 14.9443i − 0.826420i
$$328$$ 0 0
$$329$$ −6.47214 −0.356820
$$330$$ 0 0
$$331$$ −33.8885 −1.86268 −0.931341 0.364147i $$-0.881361\pi$$
−0.931341 + 0.364147i $$0.881361\pi$$
$$332$$ 0 0
$$333$$ 2.47214i 0.135472i
$$334$$ 0 0
$$335$$ 3.41641i 0.186658i
$$336$$ 0 0
$$337$$ 11.0557i 0.602244i 0.953586 + 0.301122i $$0.0973611\pi$$
−0.953586 + 0.301122i $$0.902639\pi$$
$$338$$ 0 0
$$339$$ −14.9443 −0.811661
$$340$$ 0 0
$$341$$ 0.944272 0.0511352
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 0 0
$$345$$ 8.94427 0.481543
$$346$$ 0 0
$$347$$ 30.8328i 1.65519i 0.561324 + 0.827596i $$0.310292\pi$$
−0.561324 + 0.827596i $$0.689708\pi$$
$$348$$ 0 0
$$349$$ −25.4164 −1.36051 −0.680255 0.732976i $$-0.738131\pi$$
−0.680255 + 0.732976i $$0.738131\pi$$
$$350$$ 0 0
$$351$$ −4.47214 −0.238705
$$352$$ 0 0
$$353$$ 17.5279i 0.932914i 0.884544 + 0.466457i $$0.154470\pi$$
−0.884544 + 0.466457i $$0.845530\pi$$
$$354$$ 0 0
$$355$$ −27.8885 −1.48017
$$356$$ 0 0
$$357$$ 6.47214i 0.342542i
$$358$$ 0 0
$$359$$ 13.4164 0.708091 0.354045 0.935228i $$-0.384806\pi$$
0.354045 + 0.935228i $$0.384806\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 16.8328i 0.881070i
$$366$$ 0 0
$$367$$ − 25.8885i − 1.35137i −0.737190 0.675685i $$-0.763848\pi$$
0.737190 0.675685i $$-0.236152\pi$$
$$368$$ 0 0
$$369$$ 3.52786 0.183653
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ − 16.3607i − 0.847124i −0.905867 0.423562i $$-0.860780\pi$$
0.905867 0.423562i $$-0.139220\pi$$
$$374$$ 0 0
$$375$$ 11.1803i 0.577350i
$$376$$ 0 0
$$377$$ − 37.8885i − 1.95136i
$$378$$ 0 0
$$379$$ −5.88854 −0.302474 −0.151237 0.988498i $$-0.548326\pi$$
−0.151237 + 0.988498i $$0.548326\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 20.3607i 1.04038i 0.854050 + 0.520191i $$0.174139\pi$$
−0.854050 + 0.520191i $$0.825861\pi$$
$$384$$ 0 0
$$385$$ 4.47214i 0.227921i
$$386$$ 0 0
$$387$$ − 2.47214i − 0.125666i
$$388$$ 0 0
$$389$$ 2.58359 0.130993 0.0654967 0.997853i $$-0.479137\pi$$
0.0654967 + 0.997853i $$0.479137\pi$$
$$390$$ 0 0
$$391$$ −25.8885 −1.30924
$$392$$ 0 0
$$393$$ − 13.8885i − 0.700584i
$$394$$ 0 0
$$395$$ 20.0000 1.00631
$$396$$ 0 0
$$397$$ 7.52786i 0.377813i 0.981995 + 0.188906i $$0.0604943\pi$$
−0.981995 + 0.188906i $$0.939506\pi$$
$$398$$ 0 0
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ −15.8885 −0.793436 −0.396718 0.917941i $$-0.629851\pi$$
−0.396718 + 0.917941i $$0.629851\pi$$
$$402$$ 0 0
$$403$$ − 2.11146i − 0.105179i
$$404$$ 0 0
$$405$$ 2.23607 0.111111
$$406$$ 0 0
$$407$$ − 4.94427i − 0.245078i
$$408$$ 0 0
$$409$$ 7.88854 0.390063 0.195032 0.980797i $$-0.437519\pi$$
0.195032 + 0.980797i $$0.437519\pi$$
$$410$$ 0 0
$$411$$ −19.8885 −0.981030
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 11.0557i 0.542704i
$$416$$ 0 0
$$417$$ − 2.94427i − 0.144182i
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −10.9443 −0.533391 −0.266696 0.963781i $$-0.585932\pi$$
−0.266696 + 0.963781i $$0.585932\pi$$
$$422$$ 0 0
$$423$$ − 6.47214i − 0.314686i
$$424$$ 0 0
$$425$$ − 32.3607i − 1.56972i
$$426$$ 0 0
$$427$$ − 3.52786i − 0.170725i
$$428$$ 0 0
$$429$$ 8.94427 0.431834
$$430$$ 0 0
$$431$$ 21.4164 1.03159 0.515796 0.856711i $$-0.327496\pi$$
0.515796 + 0.856711i $$0.327496\pi$$
$$432$$ 0 0
$$433$$ 21.4164i 1.02921i 0.857428 + 0.514603i $$0.172061\pi$$
−0.857428 + 0.514603i $$0.827939\pi$$
$$434$$ 0 0
$$435$$ 18.9443i 0.908308i
$$436$$ 0 0
$$437$$ − 8.00000i − 0.382692i
$$438$$ 0 0
$$439$$ 31.5279 1.50474 0.752371 0.658739i $$-0.228910\pi$$
0.752371 + 0.658739i $$0.228910\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 36.9443i 1.75528i 0.479325 + 0.877638i $$0.340882\pi$$
−0.479325 + 0.877638i $$0.659118\pi$$
$$444$$ 0 0
$$445$$ 38.9443 1.84614
$$446$$ 0 0
$$447$$ 12.4721i 0.589912i
$$448$$ 0 0
$$449$$ 9.05573 0.427366 0.213683 0.976903i $$-0.431454\pi$$
0.213683 + 0.976903i $$0.431454\pi$$
$$450$$ 0 0
$$451$$ −7.05573 −0.332241
$$452$$ 0 0
$$453$$ 17.8885i 0.840477i
$$454$$ 0 0
$$455$$ 10.0000 0.468807
$$456$$ 0 0
$$457$$ 8.94427i 0.418395i 0.977873 + 0.209198i $$0.0670852\pi$$
−0.977873 + 0.209198i $$0.932915\pi$$
$$458$$ 0 0
$$459$$ −6.47214 −0.302093
$$460$$ 0 0
$$461$$ −12.4721 −0.580885 −0.290443 0.956892i $$-0.593803\pi$$
−0.290443 + 0.956892i $$0.593803\pi$$
$$462$$ 0 0
$$463$$ 34.8328i 1.61882i 0.587245 + 0.809409i $$0.300212\pi$$
−0.587245 + 0.809409i $$0.699788\pi$$
$$464$$ 0 0
$$465$$ 1.05573i 0.0489582i
$$466$$ 0 0
$$467$$ − 19.0557i − 0.881794i −0.897558 0.440897i $$-0.854660\pi$$
0.897558 0.440897i $$-0.145340\pi$$
$$468$$ 0 0
$$469$$ −1.52786 −0.0705502
$$470$$ 0 0
$$471$$ 9.41641 0.433885
$$472$$ 0 0
$$473$$ 4.94427i 0.227338i
$$474$$ 0 0
$$475$$ 10.0000 0.458831
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −11.0557 −0.504098
$$482$$ 0 0
$$483$$ 4.00000i 0.182006i
$$484$$ 0 0
$$485$$ − 7.88854i − 0.358200i
$$486$$ 0 0
$$487$$ 13.8885i 0.629350i 0.949199 + 0.314675i $$0.101896\pi$$
−0.949199 + 0.314675i $$0.898104\pi$$
$$488$$ 0 0
$$489$$ 23.4164 1.05893
$$490$$ 0 0
$$491$$ −1.05573 −0.0476443 −0.0238222 0.999716i $$-0.507584\pi$$
−0.0238222 + 0.999716i $$0.507584\pi$$
$$492$$ 0 0
$$493$$ − 54.8328i − 2.46955i
$$494$$ 0 0
$$495$$ −4.47214 −0.201008
$$496$$ 0 0
$$497$$ − 12.4721i − 0.559452i
$$498$$ 0 0
$$499$$ −5.88854 −0.263607 −0.131804 0.991276i $$-0.542077\pi$$
−0.131804 + 0.991276i $$0.542077\pi$$
$$500$$ 0 0
$$501$$ −10.4721 −0.467861
$$502$$ 0 0
$$503$$ 39.4164i 1.75749i 0.477291 + 0.878745i $$0.341619\pi$$
−0.477291 + 0.878745i $$0.658381\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ − 7.00000i − 0.310881i
$$508$$ 0 0
$$509$$ 10.5836 0.469109 0.234555 0.972103i $$-0.424637\pi$$
0.234555 + 0.972103i $$0.424637\pi$$
$$510$$ 0 0
$$511$$ −7.52786 −0.333013
$$512$$ 0 0
$$513$$ − 2.00000i − 0.0883022i
$$514$$ 0 0
$$515$$ − 28.9443i − 1.27544i
$$516$$ 0 0
$$517$$ 12.9443i 0.569288i
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ −9.41641 −0.412540 −0.206270 0.978495i $$-0.566133\pi$$
−0.206270 + 0.978495i $$0.566133\pi$$
$$522$$ 0 0
$$523$$ 7.05573i 0.308525i 0.988030 + 0.154263i $$0.0493002\pi$$
−0.988030 + 0.154263i $$0.950700\pi$$
$$524$$ 0 0
$$525$$ −5.00000 −0.218218
$$526$$ 0 0
$$527$$ − 3.05573i − 0.133110i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 15.7771i 0.683382i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 22.9443i − 0.990118i
$$538$$ 0 0
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −22.9443 −0.986451 −0.493226 0.869901i $$-0.664182\pi$$
−0.493226 + 0.869901i $$0.664182\pi$$
$$542$$ 0 0
$$543$$ 16.4721i 0.706887i
$$544$$ 0 0
$$545$$ −33.4164 −1.43140
$$546$$ 0 0
$$547$$ 37.3050i 1.59504i 0.603289 + 0.797522i $$0.293856\pi$$
−0.603289 + 0.797522i $$0.706144\pi$$
$$548$$ 0 0
$$549$$ 3.52786 0.150566
$$550$$ 0 0
$$551$$ 16.9443 0.721850
$$552$$ 0 0
$$553$$ 8.94427i 0.380349i
$$554$$ 0 0
$$555$$ 5.52786 0.234645
$$556$$ 0 0
$$557$$ − 5.05573i − 0.214218i −0.994247 0.107109i $$-0.965841\pi$$
0.994247 0.107109i $$-0.0341594\pi$$
$$558$$ 0 0
$$559$$ 11.0557 0.467607
$$560$$ 0 0
$$561$$ 12.9443 0.546508
$$562$$ 0 0
$$563$$ − 34.8328i − 1.46803i −0.679134 0.734014i $$-0.737645\pi$$
0.679134 0.734014i $$-0.262355\pi$$
$$564$$ 0 0
$$565$$ 33.4164i 1.40584i
$$566$$ 0 0
$$567$$ 1.00000i 0.0419961i
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 10.1115 0.423151 0.211576 0.977362i $$-0.432141\pi$$
0.211576 + 0.977362i $$0.432141\pi$$
$$572$$ 0 0
$$573$$ 17.4164i 0.727581i
$$574$$ 0 0
$$575$$ − 20.0000i − 0.834058i
$$576$$ 0 0
$$577$$ − 24.4721i − 1.01879i −0.860533 0.509394i $$-0.829870\pi$$
0.860533 0.509394i $$-0.170130\pi$$
$$578$$ 0 0
$$579$$ −12.9443 −0.537946
$$580$$ 0 0
$$581$$ −4.94427 −0.205123
$$582$$ 0 0
$$583$$ 4.00000i 0.165663i
$$584$$ 0 0
$$585$$ 10.0000i 0.413449i
$$586$$ 0 0
$$587$$ − 15.0557i − 0.621416i −0.950505 0.310708i $$-0.899434\pi$$
0.950505 0.310708i $$-0.100566\pi$$
$$588$$ 0 0
$$589$$ 0.944272 0.0389080
$$590$$ 0 0
$$591$$ −19.8885 −0.818105
$$592$$ 0 0
$$593$$ − 21.3050i − 0.874890i −0.899245 0.437445i $$-0.855884\pi$$
0.899245 0.437445i $$-0.144116\pi$$
$$594$$ 0 0
$$595$$ 14.4721 0.593300
$$596$$ 0 0
$$597$$ 20.4721i 0.837869i
$$598$$ 0 0
$$599$$ 18.3607 0.750197 0.375099 0.926985i $$-0.377609\pi$$
0.375099 + 0.926985i $$0.377609\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ − 1.52786i − 0.0622194i
$$604$$ 0 0
$$605$$ −15.6525 −0.636364
$$606$$ 0 0
$$607$$ − 4.94427i − 0.200682i −0.994953 0.100341i $$-0.968007\pi$$
0.994953 0.100341i $$-0.0319933\pi$$
$$608$$ 0 0
$$609$$ −8.47214 −0.343308
$$610$$ 0 0
$$611$$ 28.9443 1.17096
$$612$$ 0 0
$$613$$ − 7.41641i − 0.299546i −0.988720 0.149773i $$-0.952146\pi$$
0.988720 0.149773i $$-0.0478543\pi$$
$$614$$ 0 0
$$615$$ − 7.88854i − 0.318097i
$$616$$ 0 0
$$617$$ 47.8885i 1.92792i 0.266047 + 0.963960i $$0.414282\pi$$
−0.266047 + 0.963960i $$0.585718\pi$$
$$618$$ 0 0
$$619$$ 11.8885 0.477841 0.238920 0.971039i $$-0.423206\pi$$
0.238920 + 0.971039i $$0.423206\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 17.4164i 0.697774i
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 4.00000i 0.159745i
$$628$$ 0 0
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −23.0557 −0.917834 −0.458917 0.888479i $$-0.651762\pi$$
−0.458917 + 0.888479i $$0.651762\pi$$
$$632$$ 0 0
$$633$$ − 25.8885i − 1.02898i
$$634$$ 0 0
$$635$$ 8.94427i 0.354943i
$$636$$ 0 0
$$637$$ 4.47214i 0.177192i
$$638$$ 0 0
$$639$$ 12.4721 0.493390
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ 7.05573i 0.278251i 0.990275 + 0.139125i $$0.0444291\pi$$
−0.990275 + 0.139125i $$0.955571\pi$$
$$644$$ 0 0
$$645$$ −5.52786 −0.217659
$$646$$ 0 0
$$647$$ − 38.2492i − 1.50373i −0.659316 0.751866i $$-0.729154\pi$$
0.659316 0.751866i $$-0.270846\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −0.472136 −0.0185045
$$652$$ 0 0
$$653$$ 1.05573i 0.0413138i 0.999787 + 0.0206569i $$0.00657577\pi$$
−0.999787 + 0.0206569i $$0.993424\pi$$
$$654$$ 0 0
$$655$$ −31.0557 −1.21345
$$656$$ 0 0
$$657$$ − 7.52786i − 0.293690i
$$658$$ 0 0
$$659$$ −12.1115 −0.471795 −0.235898 0.971778i $$-0.575803\pi$$
−0.235898 + 0.971778i $$0.575803\pi$$
$$660$$ 0 0
$$661$$ 48.2492 1.87668 0.938339 0.345717i $$-0.112364\pi$$
0.938339 + 0.345717i $$0.112364\pi$$
$$662$$ 0 0
$$663$$ − 28.9443i − 1.12410i
$$664$$ 0 0
$$665$$ 4.47214i 0.173422i
$$666$$ 0 0
$$667$$ − 33.8885i − 1.31217i
$$668$$ 0 0
$$669$$ 20.9443 0.809752
$$670$$ 0 0
$$671$$ −7.05573 −0.272383
$$672$$ 0 0
$$673$$ 21.8885i 0.843741i 0.906656 + 0.421871i $$0.138626\pi$$
−0.906656 + 0.421871i $$0.861374\pi$$
$$674$$ 0 0
$$675$$ − 5.00000i − 0.192450i
$$676$$ 0 0
$$677$$ − 2.11146i − 0.0811499i −0.999176 0.0405749i $$-0.987081\pi$$
0.999176 0.0405749i $$-0.0129189\pi$$
$$678$$ 0 0
$$679$$ 3.52786 0.135387
$$680$$ 0 0
$$681$$ 9.88854 0.378930
$$682$$ 0 0
$$683$$ 14.1115i 0.539960i 0.962866 + 0.269980i $$0.0870171\pi$$
−0.962866 + 0.269980i $$0.912983\pi$$
$$684$$ 0 0
$$685$$ 44.4721i 1.69919i
$$686$$ 0 0
$$687$$ 26.3607i 1.00572i
$$688$$ 0 0
$$689$$ 8.94427 0.340750
$$690$$ 0 0
$$691$$ 48.8328 1.85769 0.928844 0.370471i $$-0.120804\pi$$
0.928844 + 0.370471i $$0.120804\pi$$
$$692$$ 0 0
$$693$$ − 2.00000i − 0.0759737i
$$694$$ 0 0
$$695$$ −6.58359 −0.249730
$$696$$ 0 0
$$697$$ 22.8328i 0.864855i
$$698$$ 0 0
$$699$$ 2.94427 0.111363
$$700$$ 0 0
$$701$$ −0.472136 −0.0178323 −0.00891616 0.999960i $$-0.502838\pi$$
−0.00891616 + 0.999960i $$0.502838\pi$$
$$702$$ 0 0
$$703$$ − 4.94427i − 0.186477i
$$704$$ 0 0
$$705$$ −14.4721 −0.545052
$$706$$ 0 0
$$707$$ 4.47214i 0.168192i
$$708$$ 0 0
$$709$$ −35.8885 −1.34782 −0.673911 0.738812i $$-0.735387\pi$$
−0.673911 + 0.738812i $$0.735387\pi$$
$$710$$ 0 0
$$711$$ −8.94427 −0.335436
$$712$$ 0 0
$$713$$ − 1.88854i − 0.0707265i
$$714$$ 0 0
$$715$$ − 20.0000i − 0.747958i
$$716$$ 0 0
$$717$$ 15.5279i 0.579899i
$$718$$ 0 0
$$719$$ 8.94427 0.333565 0.166783 0.985994i $$-0.446662\pi$$
0.166783 + 0.985994i $$0.446662\pi$$
$$720$$ 0 0
$$721$$ 12.9443 0.482070
$$722$$ 0 0
$$723$$ − 11.8885i − 0.442140i
$$724$$ 0 0
$$725$$ 42.3607 1.57324
$$726$$ 0 0
$$727$$ − 19.0557i − 0.706738i −0.935484 0.353369i $$-0.885036\pi$$
0.935484 0.353369i $$-0.114964\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 0 0
$$733$$ − 5.63932i − 0.208293i −0.994562 0.104147i $$-0.966789\pi$$
0.994562 0.104147i $$-0.0332111\pi$$
$$734$$ 0 0
$$735$$ − 2.23607i − 0.0824786i
$$736$$ 0 0
$$737$$ 3.05573i 0.112559i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 8.94427 0.328576
$$742$$ 0 0
$$743$$ − 2.83282i − 0.103926i −0.998649 0.0519630i $$-0.983452\pi$$
0.998649 0.0519630i $$-0.0165478\pi$$
$$744$$ 0 0
$$745$$ 27.8885 1.02176
$$746$$ 0 0
$$747$$ − 4.94427i − 0.180901i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −26.8328 −0.979143 −0.489572 0.871963i $$-0.662847\pi$$
−0.489572 + 0.871963i $$0.662847\pi$$
$$752$$ 0 0
$$753$$ − 24.0000i − 0.874609i
$$754$$ 0 0
$$755$$ 40.0000 1.45575
$$756$$ 0 0
$$757$$ 30.4721i 1.10753i 0.832673 + 0.553764i $$0.186809\pi$$
−0.832673 + 0.553764i $$0.813191\pi$$
$$758$$ 0 0
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ −24.4721 −0.887114 −0.443557 0.896246i $$-0.646284\pi$$
−0.443557 + 0.896246i $$0.646284\pi$$
$$762$$ 0 0
$$763$$ − 14.9443i − 0.541019i
$$764$$ 0 0
$$765$$ 14.4721i 0.523241i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −1.05573 −0.0380705 −0.0190353 0.999819i $$-0.506059\pi$$
−0.0190353 + 0.999819i $$0.506059\pi$$
$$770$$ 0 0
$$771$$ 23.4164 0.843321
$$772$$ 0 0
$$773$$ − 16.9443i − 0.609443i −0.952441 0.304722i $$-0.901437\pi$$
0.952441 0.304722i $$-0.0985634\pi$$
$$774$$ 0 0
$$775$$ 2.36068 0.0847981
$$776$$ 0 0
$$777$$ 2.47214i 0.0886874i
$$778$$ 0 0
$$779$$ −7.05573 −0.252798
$$780$$ 0 0
$$781$$ −24.9443 −0.892576
$$782$$ 0 0
$$783$$ − 8.47214i − 0.302769i
$$784$$ 0 0
$$785$$ − 21.0557i − 0.751511i
$$786$$ 0 0
$$787$$ 36.0000i 1.28326i 0.767014 + 0.641631i $$0.221742\pi$$
−0.767014 + 0.641631i $$0.778258\pi$$
$$788$$ 0 0
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ −14.9443 −0.531357
$$792$$ 0 0
$$793$$ 15.7771i 0.560261i
$$794$$ 0 0
$$795$$ −4.47214 −0.158610
$$796$$ 0 0
$$797$$ − 28.0000i − 0.991811i −0.868377 0.495905i $$-0.834836\pi$$
0.868377 0.495905i $$-0.165164\pi$$
$$798$$ 0 0
$$799$$ 41.8885 1.48191
$$800$$ 0 0
$$801$$ −17.4164 −0.615379
$$802$$ 0 0
$$803$$ 15.0557i 0.531305i
$$804$$ 0 0
$$805$$ 8.94427 0.315244
$$806$$ 0 0
$$807$$ − 30.3607i − 1.06875i
$$808$$ 0 0
$$809$$ 24.8328 0.873075 0.436538 0.899686i $$-0.356205\pi$$
0.436538 + 0.899686i $$0.356205\pi$$
$$810$$ 0 0
$$811$$ −13.0557 −0.458449 −0.229224 0.973374i $$-0.573619\pi$$
−0.229224 + 0.973374i $$0.573619\pi$$
$$812$$ 0 0
$$813$$ − 24.4721i − 0.858275i
$$814$$ 0 0
$$815$$ − 52.3607i − 1.83411i
$$816$$ 0 0
$$817$$ 4.94427i 0.172978i
$$818$$ 0 0
$$819$$ −4.47214 −0.156269
$$820$$ 0 0
$$821$$ 19.3050 0.673747 0.336874 0.941550i $$-0.390630\pi$$
0.336874 + 0.941550i $$0.390630\pi$$
$$822$$ 0 0
$$823$$ − 15.0557i − 0.524810i −0.964958 0.262405i $$-0.915484\pi$$
0.964958 0.262405i $$-0.0845156\pi$$
$$824$$ 0 0
$$825$$ 10.0000i 0.348155i
$$826$$ 0 0
$$827$$ − 32.7214i − 1.13783i −0.822395 0.568917i $$-0.807363\pi$$
0.822395 0.568917i $$-0.192637\pi$$
$$828$$ 0 0
$$829$$ −3.52786 −0.122528 −0.0612639 0.998122i $$-0.519513\pi$$
−0.0612639 + 0.998122i $$0.519513\pi$$
$$830$$ 0 0
$$831$$ 23.4164 0.812306
$$832$$ 0 0
$$833$$ 6.47214i 0.224246i
$$834$$ 0 0
$$835$$ 23.4164i 0.810358i
$$836$$ 0 0
$$837$$ − 0.472136i − 0.0163194i
$$838$$ 0 0
$$839$$ 32.9443 1.13736 0.568681 0.822558i $$-0.307454\pi$$
0.568681 + 0.822558i $$0.307454\pi$$
$$840$$ 0 0
$$841$$ 42.7771 1.47507
$$842$$ 0 0
$$843$$ 2.94427i 0.101406i
$$844$$ 0 0
$$845$$ −15.6525 −0.538462
$$846$$ 0 0
$$847$$ − 7.00000i − 0.240523i
$$848$$ 0 0
$$849$$ 16.9443 0.581526
$$850$$ 0 0
$$851$$ −9.88854 −0.338975
$$852$$ 0 0
$$853$$ − 31.5279i − 1.07949i −0.841827 0.539747i $$-0.818520\pi$$
0.841827 0.539747i $$-0.181480\pi$$
$$854$$ 0 0
$$855$$ −4.47214 −0.152944
$$856$$ 0 0
$$857$$ 16.5836i 0.566485i 0.959048 + 0.283242i $$0.0914101\pi$$
−0.959048 + 0.283242i $$0.908590\pi$$
$$858$$ 0 0
$$859$$ −21.0557 −0.718412 −0.359206 0.933258i $$-0.616952\pi$$
−0.359206 + 0.933258i $$0.616952\pi$$
$$860$$ 0 0
$$861$$ 3.52786 0.120229
$$862$$ 0 0
$$863$$ − 18.8328i − 0.641077i −0.947236 0.320538i $$-0.896136\pi$$
0.947236 0.320538i $$-0.103864\pi$$
$$864$$ 0 0
$$865$$ 26.8328i 0.912343i
$$866$$ 0 0
$$867$$ − 24.8885i − 0.845259i
$$868$$ 0 0
$$869$$ 17.8885 0.606827
$$870$$ 0 0
$$871$$ 6.83282 0.231521
$$872$$ 0 0
$$873$$ 3.52786i 0.119400i
$$874$$ 0 0
$$875$$ 11.1803i 0.377964i
$$876$$ 0 0
$$877$$ − 49.3050i − 1.66491i −0.554093 0.832455i $$-0.686935\pi$$
0.554093 0.832455i $$-0.313065\pi$$
$$878$$ 0 0
$$879$$ −24.9443 −0.841349
$$880$$ 0 0
$$881$$ 27.5279 0.927437 0.463719 0.885983i $$-0.346515\pi$$
0.463719 + 0.885983i $$0.346515\pi$$
$$882$$ 0 0
$$883$$ 16.3607i 0.550581i 0.961361 + 0.275290i $$0.0887740\pi$$
−0.961361 + 0.275290i $$0.911226\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 4.58359i − 0.153902i −0.997035 0.0769510i $$-0.975482\pi$$
0.997035 0.0769510i $$-0.0245185\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ 12.9443i 0.433164i
$$894$$ 0 0
$$895$$ −51.3050 −1.71494
$$896$$ 0 0
$$897$$ − 17.8885i − 0.597281i
$$898$$ 0 0
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ 12.9443 0.431236
$$902$$ 0 0
$$903$$ − 2.47214i − 0.0822675i
$$904$$ 0 0
$$905$$ 36.8328 1.22436
$$906$$ 0 0
$$907$$ − 16.3607i − 0.543247i −0.962404 0.271624i $$-0.912439\pi$$
0.962404 0.271624i $$-0.0875606\pi$$
$$908$$ 0 0
$$909$$ −4.47214 −0.148331
$$910$$ 0 0
$$911$$ 25.4164 0.842083 0.421042 0.907041i $$-0.361665\pi$$
0.421042 + 0.907041i $$0.361665\pi$$
$$912$$ 0 0
$$913$$ 9.88854i 0.327263i
$$914$$ 0 0
$$915$$ − 7.88854i − 0.260787i
$$916$$ 0 0
$$917$$ − 13.8885i − 0.458640i
$$918$$ 0 0
$$919$$ −36.7214 −1.21133 −0.605663 0.795721i $$-0.707092\pi$$
−0.605663 + 0.795721i $$0.707092\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 55.7771i 1.83593i
$$924$$ 0 0
$$925$$ − 12.3607i − 0.406417i
$$926$$ 0 0
$$927$$ 12.9443i 0.425146i
$$928$$ 0 0
$$929$$ −19.3050 −0.633375 −0.316687 0.948530i $$-0.602571\pi$$
−0.316687 + 0.948530i $$0.602571\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ − 16.0000i − 0.523816i
$$934$$ 0 0
$$935$$ − 28.9443i − 0.946579i
$$936$$ 0 0
$$937$$ 15.3050i 0.499991i 0.968247 + 0.249995i $$0.0804291\pi$$
−0.968247 + 0.249995i $$0.919571\pi$$
$$938$$ 0 0
$$939$$ 25.4164 0.829433
$$940$$ 0 0
$$941$$ 35.5279 1.15818 0.579088 0.815265i $$-0.303409\pi$$
0.579088 + 0.815265i $$0.303409\pi$$
$$942$$ 0 0
$$943$$ 14.1115i 0.459532i
$$944$$ 0 0
$$945$$ 2.23607 0.0727393
$$946$$ 0 0
$$947$$ − 16.0000i − 0.519930i −0.965618 0.259965i $$-0.916289\pi$$
0.965618 0.259965i $$-0.0837111\pi$$
$$948$$ 0 0
$$949$$ 33.6656 1.09283
$$950$$ 0 0
$$951$$ 6.94427 0.225183
$$952$$ 0 0
$$953$$ 18.0000i 0.583077i 0.956559 + 0.291539i $$0.0941672\pi$$
−0.956559 + 0.291539i $$0.905833\pi$$
$$954$$ 0 0
$$955$$ 38.9443 1.26021
$$956$$ 0 0
$$957$$ 16.9443i 0.547731i
$$958$$ 0 0
$$959$$ −19.8885 −0.642235
$$960$$ 0 0
$$961$$ −30.7771 −0.992809
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 28.9443i 0.931749i
$$966$$ 0 0
$$967$$ 25.8885i 0.832519i 0.909246 + 0.416260i $$0.136659\pi$$
−0.909246 + 0.416260i $$0.863341\pi$$
$$968$$ 0 0
$$969$$ 12.9443 0.415830
$$970$$ 0 0
$$971$$ 33.8885 1.08754 0.543768 0.839236i $$-0.316997\pi$$
0.543768 + 0.839236i $$0.316997\pi$$
$$972$$ 0 0
$$973$$ − 2.94427i − 0.0943890i
$$974$$ 0 0
$$975$$ 22.3607 0.716115
$$976$$ 0 0
$$977$$ 52.8328i 1.69027i 0.534552 + 0.845136i $$0.320480\pi$$
−0.534552 + 0.845136i $$0.679520\pi$$
$$978$$ 0 0
$$979$$ 34.8328 1.11326
$$980$$ 0 0
$$981$$ 14.9443 0.477134
$$982$$ 0 0
$$983$$ 25.5279i 0.814212i 0.913381 + 0.407106i $$0.133462\pi$$
−0.913381 + 0.407106i $$0.866538\pi$$
$$984$$ 0 0
$$985$$ 44.4721i 1.41700i
$$986$$ 0 0
$$987$$ − 6.47214i − 0.206010i
$$988$$ 0 0
$$989$$ 9.88854 0.314437
$$990$$ 0 0
$$991$$ 56.9443 1.80889 0.904447 0.426586i $$-0.140284\pi$$
0.904447 + 0.426586i $$0.140284\pi$$
$$992$$ 0 0
$$993$$ − 33.8885i − 1.07542i
$$994$$ 0 0
$$995$$ 45.7771 1.45123
$$996$$ 0 0
$$997$$ 5.63932i 0.178599i 0.996005 + 0.0892995i $$0.0284628\pi$$
−0.996005 + 0.0892995i $$0.971537\pi$$
$$998$$ 0 0
$$999$$ −2.47214 −0.0782149
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1680.2.t.h.1009.4 4
3.2 odd 2 5040.2.t.u.1009.2 4
4.3 odd 2 840.2.t.c.169.2 4
5.2 odd 4 8400.2.a.db.1.1 2
5.3 odd 4 8400.2.a.cz.1.2 2
5.4 even 2 inner 1680.2.t.h.1009.2 4
12.11 even 2 2520.2.t.f.1009.1 4
15.14 odd 2 5040.2.t.u.1009.1 4
20.3 even 4 4200.2.a.bk.1.2 2
20.7 even 4 4200.2.a.bj.1.1 2
20.19 odd 2 840.2.t.c.169.4 yes 4
60.59 even 2 2520.2.t.f.1009.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.c.169.2 4 4.3 odd 2
840.2.t.c.169.4 yes 4 20.19 odd 2
1680.2.t.h.1009.2 4 5.4 even 2 inner
1680.2.t.h.1009.4 4 1.1 even 1 trivial
2520.2.t.f.1009.1 4 12.11 even 2
2520.2.t.f.1009.2 4 60.59 even 2
4200.2.a.bj.1.1 2 20.7 even 4
4200.2.a.bk.1.2 2 20.3 even 4
5040.2.t.u.1009.1 4 15.14 odd 2
5040.2.t.u.1009.2 4 3.2 odd 2
8400.2.a.cz.1.2 2 5.3 odd 4
8400.2.a.db.1.1 2 5.2 odd 4